Expanded theory of relativity: Demonstrating wave-particle duality as an integral part of relativity theory

This work develops the foundations of particle dynamics within an \emph{Expanded Theory of Relativity} that unifies special relativity with quantum wave dynamics. The framework rests on three postulates: the principle of relativity; a principle of maximal spatiotemporal symmetry; and a \emph{Regularity Postulate} requiring that the composition of two finite inertial motions yields a finite operational relative velocity. By promoting time to a three-dimensional vector subspace, the theory separates into two coupled projections: \emph{Pole dynamics}, which recovers the Lorentzian interval, and \emph{Flag dynamics}, which describes the evolution of the time vector's orientation via a unit-quaternion rotor field $\Psi$. Rather than assuming a Lorentzian signature axiomatically, we demonstrate that the temporal sign branch is fixed by enforcing the Regularity Postulate. Furthermore, since the six- and four-dimensional metrics are found to coincide along the physical worldline, the internal temporal frame must be transported without generating spurious kinetic energy. This geometric compatibility condition yields a universal covariant transport law $D_\tau \Psi = 0$. Without imposing quantization axioms, this law reproduces the Klein--Gordon, Dirac, and Proca equations depending on the chosen linear representation. In this context, $\hbar$ is identified as a geometric stiffness scale converting internal temporal rotation into external translation. The theory distinguishes itself by deriving these distinct dynamics from a single unified field equation, where particle spin arises from the algebraic carrier space rather than independent axioms.